The "day before" refers to a specific point in time from which to take the previous day.

"Yesterday" is the day before today. Not an event which may be in the past or the future but the present, today.

-- Chris.

They are actually not the same in English. 

The day before she arrived I cleaned my room. 

Yesterday I cleaned my room. 

It's the same in French. 

-- Chris. 

I am sorry my good man but yesterday is the day before 

Yesterday, before she arrived I cleaned my room

The day before, I cleaned my room

both sentences although expressed diffently say the exact same thing which is getting to the question of hier vs la veille. Is there a difference?

Chris is correct.  "Yesterday" is linked to the present, but "the day before" is not.  Indeed it can be in the past (last Saturday I went Kitesurfing, but the day before I went wakeboarding) or in the future (next Saturday I will go kitesurfing, but the day before I will go wakeboarding).